Part 10

At the same time I shall remark for which value of _c_, this invariance can be conclusively held to be true. _For c, we shall substitute the velocity of light c in free space._ In order to avoid speaking either of space or of vacuum, we may take this quantity as the ratio between the electrostatic and electro-magnetic units of electricity.

We can form an idea of the invariant character of the expression for natural laws for the group-transformation G_{_c_} in the following manner.

Out of the totality of natural phenomena, we can, by successive higher approximations, deduce a coordinate system (_x_, _y_, _z_, _t_); by means of this coordinate system, we can represent the phenomena according to definite laws. This system of reference is by no means uniquely determined by the phenomena. _We can change the system of reference in any possible manner corresponding to the above-mentioned group transformation G_{c}, but the expressions for natural laws will not be changed thereby._

For example, corresponding to the above described figure, we can call _t′_ the time, but then necessarily the space connected with it must be expressed by the manifoldness (_x′_ _y_ _z_). The physical laws are now expressed by means of _x′_, _y_, _z_, _t′_,—and the expressions are just the same as in the case of _x_, _y_, _z_, _t_. According to this, we shall have in the world, not one space, but many spaces,—quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry will be a chapter of four-dimensional physics. Now you perceive, why I said in the beginning that time and space shall reduce to mere shadows and we shall have a world complete in itself.

II

Now the question may be asked,—what circumstances lead us to these changed views about time and space, are they not in contradiction with observed phenomena, do they finally guarantee us advantages for the description of natural phenomena?

Before we enter into the discussion, a very important point must be noticed. Suppose we have individualised time and space in any manner; then a world-line parallel to the _t_-axis will correspond to a stationary point; a world-line inclined to the _t_-axis will correspond to a point moving uniformly; and a world-curve will correspond to a point moving in any manner. Let us now picture to our mind the world-line passing through any world point _x_, _y_, _z_, _t_; now if we find the world-line parallel to the radius vector OA′ of the hyperboloidal sheet, then we can introduce OA′ as a new time-axis, and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned. We shall now introduce this fundamental axiom:—

_The substance existing at any world point can always be conceived to be at rest, if we establish our time and space suitably._ The axiom denotes that in a world-point, the expression

_c²__dt²_ - _dx²_ - _dy²_ - _dz²_

shall always be positive or what is equivalent to the same thing, every velocity V should be smaller than _c_. _c_ shall therefore be the upper limit for all substantial velocities and herein lies a deep significance for the quantity _c_. At the first impression, the axiom seems to be rather unsatisfactory. It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than _c_ will play no part, something like imaginary coordinates in geometry.

The _impulse_ and real cause of inducement _for the assumption of the group-transformation G_{c}_ is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation G_{_c_}. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group G_{infinity}. Now if we have an optics with G_{_c_}, and on the other hand if there are rigid bodies, it is easy to see that a _t_-direction can be defined by the two hyperboloidal shells common to the groups G_{∞}, and G_{_c_}, which has got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orientations, with regard to the direction of progressive motion of the earth. But all efforts directed towards this object, and even the celebrated interference-experiment of Michelson have given negative results. In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis which practically amounts to an invariance of optics for the group G_{_c_}. According to Lorentz every substance shall suffer a contraction

1:(√(1 - v²/_c²_)) in length, in the direction of its motion

_l_/_l′_ = 1/√(1 - _v²_/_c²_) _l′_ = _l_(1 - _v²_/_c²_).

This hypothesis sounds rather phantastical. For the contraction is not to be thought of as a consequence of the resistance of ether, but purely as a gift from the skies, as a sort of condition always accompanying a state of motion.

I shall show in our figure that Lorentz’s hypothesis is fully equivalent to the new conceptions about time and space. Thereby it may appear more intelligible. Let us now, for the sake of simplicity, neglect (_y_, _z_) and fix our attention on a two dimensional world, in which let upright strips parallel to the _t_-axis represent a state of rest and another parallel strip inclined to the _t_-axis represent a state of uniform motion for a body, which has a constant spatial extension (see fig. 1). If OA′ is parallel to the second strip, we can take _t′_ as the _t_-axis and _x′_ as the _x_-axis, then the second body will appear to be at rest, and the first body in uniform motion. We shall now assume that the first body supposed to be at rest, has the length _l_, _i.e._, the cross section PP of the first strip upon the _x_-axis = _l_^. OC, where OC is the unit measuring rod upon the _x_-axis—and the second body also, when supposed to be at rest, has the same length _l_, this means that, the cross section Q′Q′ of the second strip has a cross-section _l_^· OC′, when measured parallel to the _x′_-axis. In these two bodies, we have now images of two Lorentz-electrons, one of which is at rest and the other moves uniformly. Now if we stick to our original coordinates, then the extension of the second electron is given by the cross section QQ of the strip belonging to it measured parallel to the _x_-axis. Now it is clear since Q′Q′ = _l_^· OC′, that QQ = _l_^· OD′.

If (_dc_/_dt_) = _v_, an easy calculation gives that

OD′ = OC √(1-(_v²_/_c²_)), therefore (PP/QQ) = (1/√(1-(_v²_/_c²_))

This is the sense of Lorentz’s hypothesis about the contraction of electrons in case of motion. On the other hand, if we conceive the second electron to be at rest, and therefore adopt the system (_x′_, _t′_,) then the cross-section P′P′ of the strip of the electron parallel to OC′ is to be regarded as its length and we shall find the first electron shortened with reference to the second in the same proportion, for it is,

P′P′/Q′Q′ = OD/OC′ = OD′/OC = QQ/PP

Lorentz called the combination _t′_ of (_t_ and _x_) as the _local time_ (_Ortszeit_) of the uniformly moving electron, and used a physical construction of this idea for a better comprehension of the contraction-hypothesis. But to perceive clearly that the time of an electron is as good as the time of any other electron, _i.e._ _t_, _t′_ are to be regarded as equivalent, has been the service of A. Einstein [Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis. ... 4-4-11-1907.] There the concept of time was shown to be completely and unambiguously established by natural phenomena. But the concept of space was not arrived at, either by Einstein or Lorentz, probably because in the case of the above-mentioned spatial transformations, where the (_x′_, _t′_) plane coincides with the _x_-_t_ plane, the significance is possible that the _x_-axis of space some-how remains conserved in its position.

We can approach the idea of space in a corresponding manner, though some may regard the attempt as rather fantastical.

According to these ideas, the word “Relativity-Postulate” which has been coined for the demands of invariance in the group G, seems to be rather inexpressive for a true understanding of the group G_{_c_}, and for further progress. Because the sense of the postulate is that the four-dimensional world is given in space and time by phenomena only, but the projection in time and space can be handled with a certain freedom, and therefore I would rather like to give to this assertion the name “_The Postulate of the Absolute world_” [World-Postulate].

III

By the world-postulate a similar treatment of the four determining quantities _x_, _y_, _z_, _t_, of a world-point is possible. Thereby the forms under which the physical laws come forth, gain in intelligibility, as I shall presently show. Above all, the idea of acceleration becomes much more striking and clear.

I shall again use the geometrical method of expression. Let us call any world-point O as a “Space-time-null-point.” The cone

_c²__t²_ - _x²_ - _y²_ - _z²_ = O

consists of two parts with O as apex, one part having _t_ < 0, the other having _t_ > 0. The first, which we may call _the fore-cone_ consists of all those points which send light towards O, the second, which we may call _the aft-cone_, consists of all those points which receive their light from O. The region bounded by the fore-cone may be called the fore-side of O, and the region bounded by the aft-cone may be called the aft-side of O. (_Vide_ fig. 2).

On the aft-side of O we have the already considered hyperboloidal shell F = _c²__t²_ - _x²_ - _y²_ - _z²_ = 1, _t_ > 0.

The region inside the two cones will be occupied by the hyperboloid of one sheet

-F = _x²_ + _y²_ + _z²_ - _c²__t²_ = _k²_,

where _k²_ can have all possible positive values. The hyperbolas which lie upon this figure with O as centre, are important for us. For the sake of clearness the individual branches of this hyperbola will be called the “_Inter-hyperbola with centre O_.” Such a hyperbolic branch, when thought of as a world-line, would represent a motion which for _t_ = -∞ and _t_ = ∞, asymptotically approaches the velocity of light _c_.

If, by way of analogy to the idea of vectors in space, we call any directed length in the manifoldness _x_, _y_, _z_, _t_ a vector, then we have to distinguish between a time-vector directed from O towards the sheet ±F = 1, _t_ > 0 and a space-vector directed from O towards the sheet -F = 1. The time-axis can be parallel to any vector of the first kind. Any world-point between the _fore_ and _aft cones_ of O, may by means of the system of reference be regarded either as synchronous with O, as well as later or earlier than O. Every world-point on the fore-side of O is necessarily always earlier, every point on the aft side of O, later than O. The limit _c_ = ∞ corresponds to a complete folding up of the wedge-shaped cross-section between the fore and aft cones in the manifoldness _t_ = 0. In the figure drawn, this cross-section has been intentionally drawn with a different breadth.

Let us decompose a vector drawn from O towards (_x_, _y_, _z_, _t_) into its components. If the directions of the two vectors are respectively the directions of the radius vector OR to one of the surfaces ±F = 1, and of a tangent RS at the point R of the surface, then the vectors shall be called normal to each other. Accordingly

_c²__tt₁_ - _xx₁_ - _yy₁_ - _zz₁_ = 0,

which is the condition that the vectors with the components (_x_, _y_, _z_, _t_) and (_x₁_ _y₁_ _z₁_ _t₁_) are normal to each other.

For the _measurement_ of vectors in different directions, the unit measuring rod is to be fixed in the following manner;—a space-like vector from 0 to -F = I is always to have the measure unity, and a time-like vector from O to +F = 1, _t_ > 0 is always to have the measure 1/_c_.

Let us now fix our attention upon the world-line of a substantive point running through the world-point (_x_, _y_, _z_, _t_); then as we follow the _progress_ of the line, the quantity

_d_τ = (1/_c_) √(_c²__dt²_ - _dx²_ - _dy²_ - _dz²_),

corresponds to the time-like vector-element (_dx_, _dy_, _dz_, _dt_).

The integral τ = ∫_d_τ, taken over the world-line from any fixed initial point P₀ to any variable final point P, may be called the “Proper-time” of the substantial point at P₀ upon the _world-line_. We may regard (_x_, _y_, _z_, _t_), _i.e._, the components of the vector OP, as functions of the “proper-time” τ; let ([._x_], [._y_], [._z_], [._t_]) denote the first differential-quotients, and ([.._x_], [.._y_], [.._z_], [.._t_]) the second differential quotients of (_x_, _y_, _z_, _t_) with regard to τ, then these may respectively be called the _Velocity-vector_, and the _Acceleration-vector_ of the substantial point at P. Now we have

_c²_ [._t²_] - [._x²_] - [._y²_] - [._z²_] = _c²_

_c²_ [._t_][.._t_] - [._x_][.._x_] - [._y_][.._y_] - [._z_][.._z_] = 0

_i.e._, the ‘_Velocity-vector_’ is the time-like vector of unit measure in the direction of the world-line at P, the ‘_Acceleration-vector_’ at P is normal to the velocity-vector at P, and is in any case, a space-like vector.

Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a ‘fore-cone’ and an ‘aft-cone.’ This hyperbola may be called the “hyperbola of curvature” at P (_vide_ fig. 3). If M be the centre of this hyperbola, then we have to deal here with an ‘Inter-hyperbola’ with centre M. Let P = measure of the vector MP, then we easily perceive that the acceleration-vector at P is _a vector of magnitude_ _c²_/ρ _in the direction of_ MP.

If [.._x_], [.._y_], [.._z_], [.._t_] are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and ρ = ∞.

IV

In order to demonstrate that the assumption of the group G_{_c_} for the physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successfully made in the case of “Thermodynamics and Radiation,”[30] for “Electromagnetic phenomena”,[31] and finally for “Mechanics with the maintenance of the idea of mass.”

For this last mentioned province of physics, the question may be asked: if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-point (_x_, _y_, _z_, _t_), where the velocity-vector is ([._x_], [._y_], [._z_], [._t_]), then how are we to regard this force when the system of reference is changed in any possible manner? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group G_{_c_} is undoubtedly permissible. These theorems lead us to the following simple rule; _if the system of reference be changed in any way, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components_

[._t_]X, [._t_]Y, [._t_]Z, [._t_]T,

_where_ T = 1/_c²_ ([._x_]/[._t_] X + [._y_]/[._t_] Y + [._z_]/[._t_] Z) = 1/_c²_ (_the rate of which work is done at the world-point_), _remains unaltered_.

This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a _moving force-vector at_ P.

Now the world-line passing through P will be described by a substantial point with the constant _mechanical mass m_. Let us call _m-times_ the velocity-vector at P as the _impulse-vector_, and _m-times_ the acceleration-vector at P as the _force-vector of motion_, at P. According to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector[32]:

_The force-vector of motion is equal to the moving force-vector._

This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the “Energy-law.” Accordingly _c²_-_times the component of the impulse-vector in the direction of the t-axis is_ to be defined as _the kinetic-energy_ of the point-mass. The expression for this is

_mc²_ _dt_/_d_τ = _mc²_ /√(1 - _v²_/_c²_)

_i.e._, if we deduct from this the additive constant _mc²_, we obtain the expression ½ _mv²_ of Newtonian-mechanics up to magnitudes of _the order of_ 1/_c²_. Hence it appears that _the energy_ depends _upon the system of reference_. But since the _t_-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting case c = ∞, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz.[33]

From the very beginning, we can establish the ratio between the units of time and space in such a manner, that the velocity of light becomes unity. If we now write √-1 _t_ = _l_, in the place of _l_, then the differential expression

_d_τ² = -(_dx²_ + _dy²_ + _dz²_ + _dl²_),

becomes symmetrical in (_x_, _y_, _r_, _l_); this symmetry then enters into each law, which does not contradict the _world-postulate_. We can clothe the “essential nature of this postulate in the mystical, but mathematically significant formula

3·10⁵ _km_ = √-1 Sec.

V

The advantages arising from the formulation of the world-postulate are illustrated by nothing so strikingly as by the expressions which tell us about the reactions exerted by a point-charge moving in any manner according to the Maxwell-Lorentz theory.

Let us conceive of the world-line of such an electron with the charge (_e_), and let us introduce upon it the “Proper-time” τ reckoned from any possible initial point. In order to obtain the field caused by the electron at any world-point P₁ let us construct the fore-cone belonging to P₁ (_vide_ fig. 4). Clearly this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from P₁ the normal to this tangent. Let _r_ be the measure of P₁Q. According to the definition of a fore-cone, _r_/_e_ is to be reckoned as the measure of PQ. Now at the world-point P₁, the vector-potential of the field excited by _e_ is represented by the vector in direction PQ, having the magnitude _e_/_cr_, in its three space components along the _x_-, _y_-, _z_-axes; the scalar-potential is represented by the component along the _t_-axis. This is the elementary law found out by A. Lienard, and E. Wiechert.[34]

If the field caused by the electron be described in the above-mentioned way, then it will appear that the division of the field into electric and magnetic forces is a relative one, and depends upon the time-axis assumed; the two forces considered together bears some analogy to the force-screw in mechanics; the analogy is, however, imperfect.

I shall now describe _the ponderomotive force which is exerted by one moving electron upon another moving electron_. Let us suppose that the world-line of a second point-electron passes through the world-point P₁. Let us determine P, Q, _r_ as before, construct the middle-point M of the hyperbola of curvature at P, and finally the normal MN upon a line through P which is parallel to QP₁. With P as the initial point, we shall establish a system of reference in the following way: the _t_-axis will be laid along PQ, the _x_-axis in the direction of QP₁. The _y_-axis in the direction of MN, then the _z_-axis is automatically determined, as it is normal to the _x_-, _y_-, _z_-axes. Let [:_x_], [:_y_], [:_z_], [:_t_] be the acceleration-vector at P, [._x_]₁, [._y_]₁ [._z_]₁, [._t_]₁ be the velocity-vector at P₁. Then the force-vector exerted by the first election _e_, (moving in any possible manner) upon the second election _e_, (likewise moving in any possible manner) at P₁ is represented by

-_e e₁_([._t₁_] - [._x₁_]/_c_)F,

_For the components F_{x}, F_{y}, F_{z}, F_{t} of the vector F the following three relations hold_:—

_c_F_{_t_} - F_{_x_} = 1/_r²_, F_{_y_} = [:_y_]/(_c²__r_), F_{_z_} = 0,

_and fourthly this vector F is normal to the velocity-vector_ P₁, _and through this circumstance alone, its dependence on this last velocity-vector arises_.

If we compare with this expression the previous formulæ[35] giving the elementary law about the ponderomotive action of moving electric charges upon each other, then we cannot but admit, that the relations which occur here reveal the inner essence of full simplicity first in four dimensions; but in three dimensions, they have very complicated projections.

In the mechanics reformed according to the world-postulate, the disharmonies which have disturbed the relations between Newtonian mechanics and modern electrodynamics automatically disappear. I shall now consider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses _m_ and _m₁_ describe their world-lines; a moving force-vector is exercised by _m_ upon _m₁_, and the expression is just the same as in the case of the electron, only we have to write +_mm₁_ instead -_ee₁_. We shall consider only the special case in which the acceleration-vector of _m_ is always zero: then _t_ may be introduced in such a manner that _m_ may be regarded as fixed, the motion of _m_ is now subjected to the moving-force vector of _m_ alone. If we now modify this given vector by writing -([^.]1/√(1-(_v_²/_c²_)) instead of [._t_] ([._t_] = 1 up to magnitudes of the order (1[^.]/_c²_)), then it appears that Kepler’s laws hold good for the position (_x₁_, _y₁_, _z₁_), of _m₁_ at any time, only in place of the time _t₁_, we have to write the proper time τ₁ of _m₁_. On the basis of this simple remark, it can be seen that the proposed law of attraction in combination with new mechanics is not less suited for the explanation of astronomical phenomena than the Newtonian law of attraction in combination with Newtonian mechanics.

Also the fundamental equations for electro-magnetic processes in moving bodies are in accordance with the world-postulate. I shall also show on a later occasion that the deduction of these equations, as taught by Lorentz, are by no means to be given up.

The fact that the world-postulate holds without exception is, I believe, the true essence of an electromagnetic picture of the world; the idea first occurred to Lorentz, its essence was first picked out by Einstein, and is now gradually fully manifest. In course of time, the mathematical consequences will be gradually deduced, and enough suggestions will be forthcoming for the experimental verification of the postulate; in this way even those, who find it uncongenial, or even painful to give up the old, time-honoured concepts, will be reconciled to the new ideas of time and space,—in the prospect that they will lead to pre-established harmony between pure mathematics and physics.

Footnote 30:

Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 1908, p. 1.

Footnote 31:

H. Minkowski; the passage refers to paper (2) of the present edition.

Footnote 32: